= Poisson extension of exponential boundary data
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{title2=$f(x,y)=\frac y\pi\int_{\mathbb R}\frac{e^{-|t|}}{(x-t)^2+y^2}\,dt$}
The displayed <Poisson integral> is the bounded <harmonic function> on the upper half-plane with continuous boundary values $f(x,0)=e^{-|x|}$. Taking a <Fourier transform> in $x$ reduces <Laplace's equation> to $\widehat f_{yy}-s^2\widehat f=0$, and the bounded solution has multiplier $e^{-y|s|}$. The <convolution theorem> gives the displayed expression. At frequency zero the transformed function is constant in $y$, not zero: its value is the total boundary integral, two.
The function decays uniformly as $x^2+y^2\to\infty$. For large $y$, use $P_y\leq1/(\pi y)$ and the boundary <L1 norm> two. When $y$ is bounded and $|x|$ is large, split the convolution into $|t|\leq |x|/2$ and its complement: the kernel is small on the first part, and the exponential boundary value is uniformly small on the second. This also distinguishes spatial decay of the harmonic function from decay of its zero-frequency Fourier coefficient.
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